C3 section 2.2 - a more sophisticated approach to functions

The approach taken in the Heinemann/Edexcel text is okay, but leaves a few subtleties untouched. For those of you wishing to study mathematics to degree-level, you ought to delve a little deeper into the definition and vocabulary of functions:


Algebraic definition of a function (derived from Stewart and Tall ^[1] )
Let X and Y be sets. A function f: X → Y is a subset of the cartesian product of the sets X × Y consisting of elements such as ( x , y ) such that:

$\displaystyle \forall \, x \in X \, \exists \, y \in Y \mid (x,y) \in f$

[for any x you choose in X, I can always find a value y in Y so that y = f(x)]

$\displaystyle \forall \, y_1 \in Y, \forall \, y_2 \in Y, \{ (x,y_1) \in f \} \wedge \{ (x,y_2) \in f \} \Rightarrow y_1 = y_2$

[if you choose a couple of 'different' values of y in Y and tell me that they are both equal to f(x), I'll tell you they weren't different to start with - this means that for all values of x, the value of f(x) is uniquely defined, a single value of y that goes with it.
Beware this does not mean that each value of x goes to a different value of y - the function f: x → 2 is a bit of a dull function, but it's perfectly legal, whereas f: x → the square root of x is not, because it could be positive or negative.]

Warning: there is some confusion and inconsistency in the use of the words "range", "codomain" and "image" in relation to set Y.
The consensus seems to be that:

• the word "codomain" describes any set Y into which the function is 'aiming'
• and the "range" is the bit of the codomain that actually gets hit when you try doing f to all possible values of x in X
• if you start with set A which is a subset of X, the bit that gets hit is called the "image" of A by f in Y.

Luckily, Edexcel and Heinemann use "range" for Y and carefully ensure that it is always the smallest set which contains f(x) for all x in X. In other words they ensure that the codomain and the range are the same, and as a result get away with never using the words 'codomain' or 'image.'

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1. ^ I. Stewart & D. Tall, The Foundations of Mathematics, Oxford Science Publications, 1997